computational issues related to sequential monte carlo filter and

Institute of Statistical MathematicsResearch Organization of Information and Systems 1

Genshiro Kitagawa Research Organization of Information and Systems

Computational Issues Related to

Sequential Monte Carlo Filter and Smoother

NUS-UTokyo Workshop on Quantitative FinanceSeptember 26, 2013


Paradigm Change of Modeling

Expansion of the object of sciences

Physical Life Human/Social Cyber-PhysicalSciences Sciences Sciences Sciences

From “science for knowledge” to “science for design”.

・From the “quest for truth” to “fulfillment of objectives (prediction, simulation, knowledge creation, decision making, control)”

・From the “physical (first principles) model” to “modeling for attaining the objectives

CPS = Cyber Physical System

Flexible platform for modeling


Outline

Sequential Monte Carlo Filter/Smoother• State-Space Model and the Kalman Filter• General State-Space Model• Sequential Monte Carlo Filter/Smoother

Computational aspects• Use of Huge Number of Particles• Two-filter Formula• Parallel MCF’s• Posterior Mean Filter/Smoother

Example• Sovereign Credit Default Swap


Ordinary State Space Model

x Fx G vy H x w

n n n

n n n

1 System ModelObservation Model

n

n

xy Time Series

State Vector n

n

wv System Noise

Observation Noise

y nx nvn

w nF

G H


State Estimation Problem

n

Prediction

Filter

Smoothing

xn

Yj

j

}, ... ,{ 1 jj yyY Observation

Obtain conditional distribution of given observationsxn Yj

Applications

• Prediction, smoothing• Likelihood, parameter estimation• Missing observations• Outliers• Signal extraction

njnjnj

for for for


Kalman Filter/Smoother

Prediction

Filter

x F xV F V F G Q G

n n n n n

n n n n n nT

n n nT

| |

| |

1 1 1

1 1 1

K V H H V H Rx x K y H xV I K H V

n n n nT

n n n nT

n

n n n n n n n n n

n n n n n n

| |

| | |

| |

( )( )

( )

1 11

1 1

1

SmoothingA V F Vx x A x xV V A V V A

n n n nT

n n

n N n n n n N n n

n N n n n n N n n nT

| |

| | | |

| | | |

( )( )

11

1 1

1 1

FilterFilter

PredictionPrediction

InitialInitial

xN, VNxN, VN

SmoothingSmoothing

n=n1

n=N

Kalman (1960)

yn

n n 1

n 1


Necessity of Non-Gaussian Modeling

Abrupt structural changes Outliers Non-Gaussian distributions

Nonlinear processes Discrete processes On-line parameter estimation

nnn vxfx )( 1Poisson processBernoulli process


Generalization of State Space Models

Nonlinear Non-Gaussiannnn

nnn

wHxyGvFxx

1

General

),(),( 1

nnn

nnn

wxhyvxfx

)| (~)| (~ 1

nn

nn

xHyxFx

Linear Gaussian

NonlinearNon-Gaussian

Discrete stateDiscrete obs.


General Recursive Filter/Smoother

Prediction

p x Y p x Y p x x p x Yp x Y

dxn N n nn n n N

n nn( | ) ( | ) ( | ) ( | )

( | )

1 1

11

p x Y p y x p x Yp y Yn n

n n n n

n n

( | ) ( | ) ( | )( | )

1

1

p x Y p x x p x Y dxn n n n n n n( | ) ( | ) ( | )

1 1 1 1 1

Filter

Smoother


Implementation of Filtering/Smoothing

• Linear-Gaussian SSM p(xn|Yk): Gaussian Kalman Filter/Smoother

• Nonlinear or Non-Gaussian p(xn|Yk): Non-Gaussian Need approximations of

non-Gaussian distributions

Prediction

Filter

)|( 1nn Yxp

)|( nn Yxp

1|1| , nnnn Vx

nnnn Vx || ,

Linear Gaussian General


Approximations of Distributions

0. Gaussian Approximation（Extended） Kalman filter/smoother

1. Piecewise-linear (Step) Approx.Non-Gaussian filter/smoother

True

Normal approx.

PiecewiseLinear

Step functionKitagawa G. JASA(1987)

Piecewise linear (or step) function approximation of the distribution and numerical integration.


Gaussian vs. Non-Gaussian Smoothing

-4

-3

-2

-1

0

1

2

3

4

1 101 201 301 401

-3

-2

-1

0

1

2

3

0 100 200 300 400

Exact Non-Gaussian Smoother

-3

-2

-1

0

1

2

3

1 101 201 301 401

Kalman Smoother

nnn

nnn

wtyvtt

1

Trend Model

Noise Distribution

　),0(~),0(~

2

2

NwNv

n

n ),0( 2Cor

Kitagawa (1987)

0 100 200 300 400 500

0 100 200 300 400 5000 100 200 300 400 500

3

2

1

0

-1

-2

-3

3

2

1

0

-1

-2

-3

4

3

2

1

0

-1

-2

-3

-4


Estimation of Volatility (Time-varying Variance)

ApproximationTrue

Model 2 AIC

Gauss 0.04318 1861.6

D-Exp 0.00011 1718.2

8

6

4

2

0

-2

-4 0 100 200 300 400 500

Gaussian Model

0 100 200 300 400 500

8

6

4

2

0

-2

-4

Cauchy Model


PTGMPPTGMK

PTTGMMKTGGMMKKMMMKKK

k

11111000300131030100300

10100110100110100130130130130

100101100101100107654321

Amount of Computation for Numerical Integration

State DimensionDimension of System NoiseNumber of NodesNodes of System Noise

jq

i

kk

m

nknkkkk mqqm

11 ~))((~

1

# of computation


Sequential Monte Carlo Methods

Gordon, Salmond and Smith (1993) Bootstrap Filter

Kitagawa (1993,1996)Monte Carlo Filter/Smoother

Doucet, de Freitas and Gordon (2001) “Sequential Monte Carlo Methods in Practice”

Frequently called “Particle Filter”


Approximations of Distributions

0. Gaussian Approximation（Extended） Kalman filter/smoother

1. Piecewise-linear (Step) Approx.Non-Gaussian filter/smoother

2. Gaussian Mixture ApproximationGaussian-sum filter/smoother

3. Particle ApproximationSeqential Monte Carlo filter/smoother

True

Normal approximation

PiecewiseLinear

Step function

Normal mixture

Particle approximation.


Approximation of Distributions by Particles

m1

Distribution function

Empirical distribution function

)|(~,, 1)()1(

nnm

nn Yxppp

F xm

I x pn nj

j

m

( ) ( ; )( )1

1

Pr( | )( )X p Ymn n

jn 1

1

)(~,,

)|(~,,

)|(~,,

)()1(

)()1(

1)()1(

nm

nn

nnm

nn

nnm

nn

vpvv

Yxpff

Yxppp

Predictive Distribution

Filter Distribution

Noise Distribution


Prediction Step

),( 1 nnn vxFx System Model

v p vnj( ) ~ ( )

f p x Ynj

n n 1 1 1( ) ~ ( | )

p F f vnj

nj

nj( ) ( ) ( )( , ) 1

11111

11111111

1111

)|()()),((

)|(),|(),,|(

)|,,()|(

nnnnnnnn

nnnnnnnnnnn

nnnnnnnn

dvdxYxpvpvxFx

dvdxYxpYxvpYvxxp

dvdxYvxxpYxp

)(~

)|(~)(

11)(

1

nj

n

nnj

n

vpv

Yxpf

p F f v p x Ynj

nj

nj

n n( ) ( ) ( )( , ) ~ ( | ) 1 1


Filtering Step (Resampling)

:)( jn Importance weight of particle pn

j( )

nj

n n njp y X p( ) ( )( | )

Pr( | ) Pr( | , )Pr( | ) Pr( | )

Pr( | ) Pr( | )

( ) ( )

( ) ( )

( ) ( )

( )

( )

( )

( )

X p Y X p Y yy X p X p Y

y X p X p Y

n nj

n n nj

n n

n n nj

n nj

n

i

m

n n ni

n ni

n

nj

m

nj

i

mm

nj

nj

i

m

1

1

11

1

11

1

nj( )

yn

p y pn nj( | )( )

pnj( )

f nj( )

f nj( )

m1

)( jn

resampling


One Cycle of Monte Carlo Filtering

)|( 11 nn Yxp

n=n+1

xSt

ate

spac

e

)|( 1nn Yxp

Resampling

)|( 1nn Yxp )|( nn Yxp

Filtering

Importance weight

X perish

X perish

)|( )()( jnn

jn xyr)( nvp

Prediction

),( )(1

)(n

jn

jn vxfx

new data yn


Computational Cost of Filtering

Sequential filtering: O(n)n: data length

Numerical integration: O(nkl+1)l: dimension of the statek: number of nodes (>100)

MCF: O(nm) , O(nmlog(m)), O(nm2)m: number of particles

m does not increase so rapidly as kl+1


Accuracy in Computing Log-Likelihood

Gaussian model Cauchy modelm Log-L S.D. CPU-time Log-L S.D. CPU-time

102 -750.859 2.287 0.02 -752.207 6.247 0.02

103 -748.529 1.115 0.06 -743.244 2.055 0.06

104 -748.127 0.577 0.58 -742.086 0.429 0.63

105 -747.960 0.232 5.84 -742.024 0.124 6.27

106 -747.931 0.059 59.41 -742.029 0.038 62.73

107 -747.926 0.023 591.04 -742.026 0.013 680.33

108 -747.930 0.008 5906.62 -742.026 0.003 6801.55

109 -747.928 0.002 59077.35 -742.026 0.001 69255.03


Fixed-lag Smoothing (by Storing Particles)

Filter p nj( ) f n

j( )

Fixed-interval Smoothing

L-lag Smoothing

s s pnj

n nj

nj

1 1 1 1|( )

|( ) ( ), , , s sn

jn n

j1 |( )

|( ), ,

s s pn L nj

n nj

nj

|( )

|( ) ( ), , ,1 1 1 s sn L n

jn n

j |

( )|

( ), ,

L particles


Number of Different Particles in Fixed-Lag Smoothing

1

10

100

1000

10000

100000

1000000

0 25 50 75 100 125 150 175 200

M=100 M=1000 M=10000M=10^5 M=10^6

1

10

100

1000

10000

100000

1000000

0 25 50 75 100 125 150 175 200

M=100 M=1000 M=10000M=10^5 M=10^6

Gaussian model Cauchy model

Lag Lag

m=100m=105

m=1000m=106

m=10000 m=10000m=1000m=106

m=100m=105


Single MCF

-3

-2

-1

0

1

2

3

0 100 200 300 400

-3

-2

-1

0

1

2

3

1 101 201 301 401

m=100,000 m=10,000

Exact Non-Gaussian Smoother

-3

-2

-1

0

1

2

3

1 101 201 301 401

-3

-2

-1

0

1

2

3

1 101 201 301 401

m=100

m=1,000

-3

-2

-1

0

1

2

3

1 101 201 301 401

m=10,000


Accuracy vs. Number of Particles

0.0001

0.001

0.01

0.1

1

10

100

1 2 3 4 5 60.001

0.01

0.1

1

10

100

1 2 3 4 5 6

Filter Smoother Fixed-interval

102 103 104 105 106 107 102 103 104 105 106 107

Number of Particles Number of Particles


Smoothing (max lag) Filter xnxDnxDDDI

N

n

L

jjj

2

1 1),(),(),(

Smoothing (best lag)


Accuracy of Fixed-Lag Smoother


m=100

m=103

m=107

m=106

m=105

m=104

m: Number of particles


Accuracy of Fixed-Lag Smoother


m=100 m=103 m=105 m=106m=104 m=107

m=100

m=103

m=104

m=105

m=106

m=107


Summary of MCF (Particle Filter/Smoother)

• Very flexible and easy-to-implement method for nonlinear non-Gaussian time series modeling.

• Due to the collapsing caused by repeated re-sampling, it is difficult to get precise posterior distribution (in particular, the quantile or percentile points), for small number of particles.

• This collapsing of smoothed distribution can be mitigated by using huge number of particles (at the expense of computational cost).


)|()|(

)|,(

),|()|(

1

1

1

nnnn

N

nn

Nn

nNnnNn

YxpxYp

YYxp

YYxpYxp

Smoothing by Two Filter Formula

Smoother

Filter

)|()|(

)|,(

),|()|(

1

1

1

nnnn

nnn

nnnnn

Yxpxyp

Yyxp

yYxpYxp Nn

nN

nn

yyY

yyY

,,

,,1

)|(

)|(

nn

N

nn

xYp

xyp

nn

n yY


Backward Filtering

)|()|( NNNN

N xypxYp

Backward Filtering

)|()|()|(

)|()|()|(1

11111

nn

Nnnnn

N

nnnnn

Nnn

N

xYpxypxYp

dxxxpxYpxYp

Initialization


Fixed-Lag with LAG=500 Two-Filter Formula

m=100,000

32

m=10,000

m=1,000


mGaussian Model Cauchy Model

Fixed-lag

Fixed-interval

Two-filter

Fixed-lag

Fixed-interval

Two-filter

102 8.693 41.723 6.913 21.248 47.881 26.440103 2.259 16.275 1.399 6.042 23.654 4.870104 0.717 5.547 0.333 1.001 3.679 0.378105 0.185 1.448 0.118 0.140 0.380 0.072

Accuracy of Smoothing


Two-Filter Formula for Smoothing

Backward FilteringFiltering

)|( nn

N xYp

Smoothing

n-1n-2 n+2n+1n


Accuracy of Two-Filter Smoothing Algorithm


1 10 102 103 104

Acc

urac

y

Acc

urac

y

ms: Number of evaluated particles ms: Number of evaluated particles1 10 102 103 104

ms=100 is sufficient.ms=10 might be reasonable


Summary of Two-Filter Formula for Smoothing

• Collapse of smoothing distribution by resampling can be mitigated by the two-filter formula.

• Strict realization of the formula is time-consuming for large m.

• Thinning (using only 10-100 particles for evaluating the particles) yields reasonable results.


Application to High-dim. Problems

Accuracy ~ O(m-1/2)

but amount of computation increases

Parallel computation

High-dimensional cases:• NGF(Integration): Impossible• MCF(Monte Carlo): Possible


0.001

1

1000

1000000

1E+09

1E+12

1E+15

1960 1970 1980 1990 2000 2010 20200.001

1

1000

1000000

1E+09

1E+12

1E+15

1960 1970 1980 1990 2000 2010 2020

History of Computers in ISM

Eflops

Pflops

Tflops

GFlops

MFlops

KIPS

EB

PB

TB

GB

MB

KB

Speed Memory

K computer K computer

NGF MCF

Parallel processor

ISM = Institute of Statistical Mathematics, Japan


Parallel Computation

Direct Parallel Implementation of MCF Simple Parallel MCF

• Simply run k MCF’s in parallel • Take an average of posteriors

Weighted Parallel MCF• Weighted average of posteriors

Weighted parallel MCF with Crossover• Crossover between MCFs, for MCF with low weight.


Direct Parallel Implementation of MCF

)()1( ,, knn ff

)2()1( ,, kn

kn ff )()( ,, m

nkm

n ff

)()1( ,, knn pp )2()1( ,, k

nk

n pp )()( ,, mn

kmn pp

Local sum of

Total sum of )( jn

)( jn

)(0

)1(0 ,, mff

)(0

)1(0 ,, kff )2(

0)1(

0 ,, kk ff )(0

)1(0 ,, mkm ff Initialization

Prediction

Filter


Communication Between Cores

1. Likelihood for j=1,…,m; i=1,…,k 2. Importance weight of i-th MCF

for i=1,…,k

3. Cumulative weight

for i=1,…,k

4. Starting points

),( ijn

m

jij

ni

n 1),()(

k

ji

n

i

ji

nin

1)(

1)(

)(

kiJ

kipI

q

pIp

nIi

n

pin

ii

1),()1(

n)(

)(n

)(

)(1

)(1 s.t. qsmallest the:

s.t. smallest the:

0

0.002

0.004

0.006

0.008

0.01

0.012

4 8 16 32

Gauss Cauchy

Number of CoresR

atio

of C

omm

unic

atio

ns B

etw

een

Cor

es

Stratified search


Efficiency of Parallel Computation

0

0.2

0.4

0.6

0.8

1

1 2 3 4 5 6 7 8 9

4-Core 8-Core16-Core 32-Core64-Core 128-Core

0.01

0.1

1

10

100

1000

10000

100000

1000000

1 2 3 4 5 6 7 8 9

1-Core 4-Core 8-Core16-Core 32-Core 64-Core128-Core

102 103 104 105 106 107 108 109 1010 102 103 104 105 106 107 108 109 1010

Number of Particles Number of Particles

Elapsed Time Relative Efficiency

Amdahl’s lawParallel portion of thealgorithm: 0.989-0.980

),(),1(),(REmkTk

mTmk

),( mkT

m


Accuracy of Simple Parallel MCF

0.00001

0.0001

0.001

0.01

0.1

1

10

1 10 100 1000

Number of Parallel MCF Number of Parallel MCF

m=102 m=103 m=105 m=106m=104

0.00001

0.0001

0.001

0.01

0.1

1

10

100

1 10 100 1000

Cauchy ModelGaussian Model



Number of parallel MCF, k Number of parallel MCF, k

m 1 10 100 100 1 10 100 1000

102 3.22273 1.11807 0.90401 0.90421 21.72463 11.32517 10.44000 10.26888

103 0.56848 0.23308 0.18685 0.18304 4.01454 1.01333 0.72036 0.72802

104 0.11893 0.02979 0.02150 0.01916 0.37586 0.03984 0.00677 0.00479

105 0.02650 0.00377 0.00141 0.00081 0.03416 0.00327 0.00034 0.00008

106 0.00396 0.00042 0.00004 0.00334 0.00032 0.00004

107 0.00039 0.00004 0.00031 0.00003

108 0.00017 0.00003




m 100 1,000 10,000 100,000

GaussianBias(m) 0.9109 0.1883 0.0200 0.0011

Variance(m) 2.2741 0.3329 0.0931 0.0240

CauchyBias(m) 10.2735 0.6985 0.0034 0.0000

Variance(m) 10.9688 3.3143 0.3724 0.0342

)(variance1)(bias )(Accuracy mk

mD

k: number of parallel MCF


Weighted Parallel MCF

MCP1

MCF1

MCP2

MCF2

MCPk

MCFk

WP-MCF

・・・

・・・

m

j

jnm 1

)1,(1

1

m

j

jnm 1

)2,(2

1

m

j

kjnk m 1

),(1

k

i i

iiw

1

k

iNniiNn YxpwYxp

1)|()|(


• Weighted parallel MCF surrogates the MCF with particles.

• However, weights of some MCF’s may become very small and that will eventually deteriorate the efficiency of the total filter.

• To alleviate this problem, perform the following crossover between MCFs

If , for j=2,4,…,m, interchange particles and weights

Weighted Parallel MCF with Crossover

),(),(

),(),(

maxmin

maxmin

ijn

ijn

ijn

ijn pp

m

j

ijn

k

iinn

ik

iinn w

mYypwYyp

1

),(

11

)(

11

1)|()|(

Cww

i

i minmax

km


100 parallel MCF

-3

-2

-1

0

1

2

3

1 101 201 301 401

-3

-2

-1

0

1

2

3

1 101 201 301 401

-3

-2

-1

0

1

2

3

1 101 201 301 401

Weighted Parallel MCF（with crossover）

-3

-2

-1

0

1

2

3

1 101 201 301 401

10 parallel MCF

-3

-2

-1

0

1

2

3

1 101 201 301 401

-3

-2

-1

0

1

2

3

1 101 201 301 401

m=1,000

m=10,000

m=100,000

-3

-2

-1

0

1

2

3

1 101 201 301 401

-3

-2

-1

0

1

2

3

1 101 201 301 401

-3

-2

-1

0

1

2

3

1 101 201 301 401

1000 parallel MCF


Simple parallel Weighted Parallel

0.00001

0.0001

0.001

0.01

0.1

1

10

100

1 10 100 1000

Accuracy (Divergence from “True” Densities)

MCF with m=100MCF with m=1,000MCF with m=10,000MCF with m=100,000MCF with m=1,000,000

D(q(x);p(x))q(x): “true” distributionp(x): estimated distribution

k : number of parallel MCF’s

D(q

;p)

m: number of particles in each MCF

?


Summary of Parallel Computation

• In parallel computation, reduction of communications between cores is crucial and stratified sampling is necessary. Parallel computation with 64 (or more) cores is efficient only for the number of particles m is 106 or more.

• Simple parallel MCF is perfectly computationally efficient but increase of the accuracy is limited for m=104 or smaller. This is because bias of the MCF cannot be reduced by simple parallel MCF.

• Weighted parallel MCF is efficient both computationally and in accuracy.


Posterior Mean Filter/Smoother

1

1)(1)(

)(1|

)1(1|

)(|

)()1()(

)()1(

)( re whe

)1( )iii(

)1( )i(

)0(

in

in

int

int

jnt

in

in

jn

mnn

u

sss

ppf

pp

MCF PMF

1

1)(1)(

)(1|

)(|

)()(|

)()(1

)(1)(1

1)(1

1)(

)(

)()(

)()()(

)(

)( re whe

)iii(

)ii(

(i)

)1,0(~ )4(

)|( )3(

) ,( )2(

)(~ )1(

in

in

jnt

jnt

jn

jnn

in

jn

in

jn

in

mn

jn

jnn

jn

jn

jn

jn

jn

u

ss

fs

pf

cuc

c

Uu

pyp

vfFp

vqv


Posterior Mean Filter/Smoother (Gauss)

-3

-2

-1

0

1

2

3

1 101 201 301 401

-2

-1

0

1

2

1 101 201 301 401

-2

-1

0

1

2

0 100 200 300 400

MCF PMF

-2

-1

0

1

450 460 470 480 490


Posterior Mean Filter/Smoother (Cauchy)

-3

-2

-1

0

1

2

3

1 101 201 301 401

MCF PMF

-2

-1

0

1

2

1 101 201 301 401

-2

-1

0

1

2

0 100 200 300 400

-1

0

1

450 460 470 480 490


-2

-1

0

1

2

3

0 10 20 30 40 50

-2

-1

0

1

2

0 10 20 30 40 50

-2

-1

0

1

2

0 10 20 30 40 50


5.0

random },{1,

mj 5.0 ,1 jj

(0,1)~

random },{1,

U

mj

i.i.d. case

-2

-1

0

1

2

3

0 10 20 30 40 50

data ordered5.0 ,1 jj



2/)ZZ( )(1

)(1

)( jjn

in

jnZ

)( Z,ZCov ,ZVar ,ZE )()()()( kjcv nk

nj

nnj

nnj

n

0 , , 02

00 cv

11

11

431

43

112111

21

nnn

nnn

cm

vm

c

cm

vm

v

21

21

41

213 3 )12(

41

21)1(23 )12(

n

n

n

n

mmc

mmmv

i.i.d. case: average

123limlim

m

cv nnnn



(0,1)~ ,ZZ)1( )(1

)(1

)( UZ jjn

in

jn

)( Z,ZCov ,ZVar ,ZE )()()()( kjcv nk

nj

nnj

nnj

n

0 , , 02

00 cv

11

11

431

43

31

32

nnn

nnn

cm

vm

c

cvv

21

21

43

323 9 )94(

43

3249 )94(

n

n

n

n

mmc

mmmv

i.i.d. case: random mean

949limlim

m

cv nnnn


Posterior Mean vs. Fixed-Lag SmootherCauchy modelGaussian model

m=102 m=103 m=105 m=106m=104m=102 m=103 m=105 m=106m=104


Summary of Posterior Mean Smoother

• Posterior mean smoother can provide only the posterior mean.

• But its accuracy is significantly higher than the fixed-lag smoother for any lag length.


Applications of MCF

Gordon et al. (1993), Kitagawa (1996)Doucet, de Freitas and Gordon (2001) “Sequential Monte Carlo Methods in Practice”

1. Non-Gaussian smoothingLevel shiftNon-Gaussian seasonal adjustmentStochastic volatility models

2. Nonlinear smoothingTrackingPhase-unwrapping

3. Signal extraction problems4. Modeling count data5. Self-organizing state space model6. High-dimensional filtering/smoothing


SovCDS Index and Analysis of Relationship between Regional Sovereign Risk the Power

Contribution.

Presented at the 59th World Statistics Congress, Hong Kong, China onAugust 26, 2013

Yoko Tanokura Meiji University, TokyoHiroshi Tsuda Doshisha University, KyotoSeisho Sato The University of Tokyo, TokyoGenshiro Kitagawa ROIS, Tokyo


Outline of the Example

Introduce Sovereign Credit Default Swap (SovCDS) and problems on SovCDS data for constructing a market index as a proxy of sovereign risk

Propose a method of index construction based on time series analysis

Construct five regional sovereign risk indices Investigate the relationships between regional sovereign risks by

using power contribution analysis and detect spillover effects of the European sovereign debt crisis


CDSbuyer

CDSseller

periodically pay a CDS spread quoted as annual rate

make a payment on the occurrence of the credit event

Sovereign Credit Default Swap (SovCDS)

...is an insurance contract that protects the buyer against the issuer’s credit risk of the country’s debt.

... can be regarded as the market evaluation on the credit risk for the country’s economy.

Daily composite spreads (provided by Markit) 82 US dollar-denominated sovereign CDS 5-year issues September 11, 2003 – March 29, 2013

Construct a Sovereign Risk Index Based on SovCDS Price Data


Problems: Heavy-tailed distributions Time-varying number of observations

Time Series of SovCDS Price Distributions

3/12/09

3/8/12

0% 51%

3/29/13

97% 144% 191% 232%

11/18/11

7/9/10

2/27/09

10/19/07

6/9/06

1/28/05

9/19/03

Number of observations


Method of Market Index Construction*

is estimated by a time-varying variance model (Kitagawa 1987).

0)(log0)1)((

))(()(1

,

npnp

nphnqi

iii

Apply Box-Cox transformation (Box and Cox 1964) to the prices

Determine an optimal λ by minimizing : modified (Akaike 1973) to the original prices (Kitagawa 2010)

)()()()()1()(

nwnxHnynvGnxFnx

))()(,0(~)(),0(~)(

),()()(),(

2

2

nknNnwNnv

nwntnynvtl

)1()()( ntntnt

Obtain the posterior distribution by applying state-space modeling (Kitagawa 2010)

For each λ, fit the following trend model to the mean time series of

* Improved version of Tanokura et al. (2012)

)(1

log2AIC'AICnzzdz

dhT

n

))(()(

Jacobian:1 nyhnz

dzdh

An index is defined by the inverse Box-Cox transformation of the optimal trend.

)(ny

2)(n

'AIC

nsobservatio ofnumber :)(,...,1time:,...,1

nkiTn

AIC

)(, nqi

1

2

3

4

5


Countries of Five Regions

LAAP EE

MADE

SovCDS Distributions


AIC-Optimal Transformation

For all regions, we select , which is the reciprocal square root transformation. 5.0

'AIC for


Examples of the Estimated Trend

Asia Pacific

Developed Europe


Asia Pacific (AP) Index & price distributions

Developed Europe (DE) Index & price distributions

Sovereign Risk Index and the Price Distributions


Five Regional Sovereign Risk Indices

RISK


Relationships between Cyclical Fluctuations around the Trends

Cyclical component: extracted by the program package Web DECOMP developed by ISM (Gersch and Kitagawa 1983, Kitagawa and Gersch 1984)

Post-Greece: 11/17/09 -3/8/12Post-Lehman: 9/15/08 -11/16/09 Current: 3/12/12 -3/29/13


Multivariate AR Model and Power Contributions(PC)

nmn

M

mm vxAx n

1 )( )( ),( )(

,)( ,)(

mnOxvEmnOvvE

WvvEOvE

T

T

T

mn

mn

nnn

nxmA

: 5-dim stationary time series: AR coefficient matrix: 5-dim white noise: Variance covariance matrix

nvW

PC measures the influence between variable fluctuations of the noise at a frequency.

Power spectrum (PS) of A:decomposes the fluctuation by frequency

PC of A: decomposes PS of A into components of variable combinations

(Akaike 1968, Tanokura and Kitagawa 2004)


Power Contributions (%)

Post-Greece: 11/17/09 -3/8/12

Post-Lehman: 9/15/08 -11/16/09

Current: 3/12/12 -3/29/13

Asia Pacific (AP) Dev. Europe (DE) Emerg. Europe (EE)Mid. East/Africa (MA) Latin America (LA)


Contribution Score(CS)

...is defined as the quantity (% of the total) between two variables based on the sum of the equally allocated the PC value to variables concerned at the dominant frequency domain of the power spectrum each region.

CS from Developed Europe can be regarded as the influence of the European debt crisis. CSs from Developed Europe for the current period become higher!


Contribution Score (Cont.)The mutual strong contribution scores between Asia Pacific and Emerging Europe are detected for the Post-Greece period.


Summary of this Example

We presented a method constructing an index where the price distributions are heavy-tailed in the market.

We showed the effectiveness of our method by applying to the sovereign Credit Default Swap (SovCDS) market.

We detect the worldwide spillover effects of the European debt crisis.

Applying our method to the markets with insufficient information such as fast-growing or immature markets can be effective.


Summary

• General state-space model and sequential Monte Carlo filter/smoother.

• Computational aspect of sequential MonteCarlo method are presented

• MCF with huge number of particles• Two-filter formula• Parallel MCF• Posterior mean smoother

• Application: Sovereign risk index

computational issues related to sequential monte carlo filter and

Documents